How to apply the ratio test to determine if Sigma (n!)/n^n from n=[1,oo) is convergent to divergent?

Hailey Oneill

Hailey Oneill

Answered question

2023-03-20

How to apply the ratio test to determine if Σ n ! n n from n = [ 1 , ) is convergent to divergent?

Answer & Explanation

cerakank2n

cerakank2n

Beginner2023-03-21Added 6 answers

n = 1 n ! n n from n = [ 1 , )
| a n + 1 a n | = ( n + 1 ) ! ( n + 1 ) n + 1 n ! n n = n n ( n + 1 ) n + 1 ( n + 1 ) ! n ! = 1 n + 1 ( n n + 1 ) n ( n + 1 ) = ( n n + 1 ) n
( n n + 1 ) n = ( 1 n + 1 n ) n = 1 ( 1 + 1 n ) n
lim n | a n + 1 a n | = 1 ( 1 + 1 n ) n = 1 e < 1

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